An ordered field is an algebraic structure like the field of real numbers. However, while the field of real numbers have only one ordering, an arbitrary ordered field F may have more than one ordering, and also a infinite and uncountble number of orderings is allowed. To each element x Î F one can associate an binary quadratic form [1, x], called Pfister 1-fold form. The set of elements in F = F 0} which are represented by [1, x] is a group D[1,x], called value group of [1,x]. An element d Σ F is called rigid if D[1, d] = F2 U dF2, where F 2 denotes the subgroup of squares in F . An element d is called birigid if d and -d are both rigid. The main purpose of this thesis is to prove an structure theorem for Witt ring (of equivalence classes of quadratic forms) of an ordered field F with a rigid element which is not birigid and is negative in at least one ordering of F, that is, we get a decomposition of the Witt ring of F as a product of Witt rings of extensions H  F and K  F, both inside the quadratic closure of F. The Witt rings of H and K have a simpler structure than Witt ring of F. We get fields H and K by builting subgroups Rd and Sd associated to the rigid element d and making the addicional assumption that F = Rd·Sd holds. The field H is a henselization of F relative to a valuation ring (A;mA) of F such that Rd = (1 + mA) F2. The pythagorean field K has space of orderings XK homeomorphic to X/Sd, the space of orderings of F which contain Sd. Moreover, we settle an necessary and suficient condiction to decomposition F = Rd·Sd holds, relative to value group and residue field of valuation ring A.

In this work we consider the Galois point of view in determining the structure ofa space of orderings of fields via considering small Galois quotients of absolute Galoisgroups G F of Pythagorean formally real fields. Galois theoretic, group theoretic andcombinatorial arguments are used to reduce the structure of W-groups.

…property −1 ∈
/
7
F 2 a formallyreal
field. They characterized formallyrealfields as fields… …Pythagorean formallyrealfields
F . Two extreme cases for F have been studied in the literature… …and T.T ⊆ T .
F
For Pythagorean formallyrealfields it is easy to check that
˙2
F 2 = F 2… …more details in Chapter 2.
√
˙ ) as
Now for a Pythagorean formallyreal field F , define… …Pythagorean formallyreal field F such that the number of
˙ /F˙ 2 is finite, the W-group of field F…